You are shown three boxes, and told that one of them contains a prize. You are then asked to pick one box, and if that box is the one with the prize, you will win it. After picking a box, you are shown that one of the other two boxes is empty, and offered a chance to change your selection.

Should you do this? Would changing your choice to the other remaining box affect your odds of winning? Why or why not?

The answer is clearly that you should switch.  I've seen an alternate explanation that may help those who still disagree, it's a little more intuitive, although no less reliant on the mathematics.

Let's say there are initially 1,000 doors.  One of them conceals a prize, and the other 999 doors have a goat.  Initially you choose a door (we'll call it door #1 for simplicity, but obviously it doesn't matter what door you open). 

After choosing your door, the host (who knows where the prize is) opens up 998 of the other doors that he knows do not contain the prize and then offers you the opportunity to switch.  Now there are only two doors left - door #1, which you initially chose, and another door (call it door #673) which the host left closed.  What do you do?

There are only two doors left, but do you really think the odds are 50/50 now?  Do you really think that when you chose your initial door out of the 1,000, there was a 50% chance that you were right?  Or does it seem far more likely that you initially chose the wrong door and the prize is behind the only other door the host intentionally left closed?  Your intuition should tell you that the prize is most likely behind the other door, and the numbers back it up - there is a 0.1% chance that the prize is behind the door you initially chose, and there is a 99.9% chance that it is behind door #673.  (If you don't believe it, try this with a friend and see how often you win when you stick with the door you originally chose!)

This is the same exact problem, just with more doors. 

Posted by tomarken on 2009-03-25 11:53:52